Free Access
Volume 494, Number 2, February I 2009
Page(s) 623 - 636
Section Interstellar and circumstellar matter
Published online 20 November 2008

Online Material

Appendix A: The reaction rate table used in the Nahoon modified chemical model

Table A.1:   Reactions used in the model. The rate coefficients are given for 7 K. Reaction rates less than 10-15 cm-3 s-1 are not taken into account in our models. Reference (1) corresponds to Gerlich (1990), references (2)-(4) correspond to Hugo, OSU 07, and this paper respectively. For OSU 07, branching ratios involving spin states have been infered from quantum mechanical rules. For reactions involving grains, a grain radius of 0.1 $\mu $m and a sticking coefficient of 1 have been considered. (5) Datz et al. (1995b) (6) Datz et al. (1995a), (7) Zhaunerchyk et al. (2008), (8) Larsson et al. (1997), (9) Molek et al. (2007).

Appendix B: On the rate coefficients for dissociation recombination of H3+ and its isotopologues

The dissociative recombination (DR) rate coefficients for ortho- and para-H3+ have been published by Fonseca dos Santos et al. (2007). Here, we present the results obtained for all four H3+ isotopologues. The DR rate coefficients for different species of the nuclear spin are calculated using the approach described in a series of papers devoted to DR theory for triatomic molecular ions. See Fonseca dos Santos et al. (2007); Kokoouline & Greene (2003a,b) for H3+ and D3+ calculations and Kokoouline & Greene (2004,2005) for H2D+ and D2H+. The scope of this paper does not allow us to review the theoretical approach in detail. We only list its main ingredients.

The theoretical approach is fully quantum mechanical and incorporates no adjustable parameters. It relies on ab initio calculations of potential surfaces for the ground electronic state of the H3+ ion and several excited states of the neutral molecule H3, performed by Mistrik et al. (2001).

The total wave function of the system is constructed by an appropriate symmetrization of products of vibrational, rotational, electronic, and nuclear spin factors. Therefore, rovibronic and nuclear spin degrees of freedom are explicitly taken into account.

The electronic Born-Oppenheimer potentials for the four H3+ (and H3) isotopologues have the C3v symmetry group. The C3v symmetry group has a two-dimensional irreducible representation E. The ion has a closed electronic shell. The lowest electronic state of the outer electron in H3 has p-wave character. The p-wave state of the electron also belongs to the E representation. Due to the Jahn-Teller theorem (Landau & Lifshitz 2003), this leads to a strong non-adiabatic coupling between the E-degenerate vibrational modes of the ion and the p-wave states of the incident electron. The coupling is responsible for the fast DR rate (Kokoouline et al. 2001) in H3+. In the present model, only the p-wave electronic states are included because other partial waves have a much smaller effect on the DR probability: the s-wave states have no E-type character and, therefore, are only weakly coupled to the dissociative electronic states of H3; d-wave electronic states are coupled to the E-vibrational modes, but the coupling is rather small because the d-wave of the incident electron does not penetrate sufficiently close to the ionic core owing to the d-wave centrifugal potential barrier.

All three internal vibrational coordinates are taken into account. Vibrational dynamics of the ionic core are described using the hyper-spherical coordinates, which represent the three vibrational degrees of freedom by a hyperradius and two hyperangles. The hyperradius is treated as a dissociation coordinate that represents uniformly the two possible DR channels, the three-body (such as H+H+H) and two-body (such as H2+H). Although the initial vibrational state of the ion is the ground state, after recombination with the electron, other vibrational states of the ionic target molecule can be populated. Therefore, in general, many vibrational states have to be included in the treatment. In particular, the states of the vibrational continuum have to be included, because only such states can lead to the dissociation of the neutral molecule. The vibrational states of the continuum are obtained using a complex absorbing potential placed at a large hyperradius to absorb the flux of the outgoing dissociative wave.

Since the rovibrational symmetry is D3h for H3+ and D3+ and C2v for H2D+ and D2H+, the rovibrational functions are classified according to the irreducible representations of the corresponding symmetry groups, i.e. A1', A1'', A2', A2'', E', and E'' for D3h and A1, A2, B1, and B2 for C2v. We use the rigid rotor approximation, i.e. the vibrational and rotational parts of the total wave function are calculated independently by diagonalizing the corresponding Hamiltonians. In our approach, the rotational wave functions must be obtained separately for the ions and for the neutral molecules. They are constructed in a different way for the D3h and C2v cases. The rotational eigenstates and eigenenergies of the D3h molecules are symmetric top wave functions (see, for example Bunker & Jensen 1998). They can be obtained analytically if the rotational constants are known. The rotational constants are obtained numerically from vibrational wave functions, i.e. they are calculated separately for each vibrational level of the target molecule. The rotational functions for the C2v ions are obtained numerically by diagonalizing the asymmetric top Hamiltonian (Kokoouline & Greene 2005; Bunker & Jensen 1998).

Once the rovibrational wave functions are calculated, we construct the electron-ion scattering matrix (S-matrix). The S-matrix is calculated in the framework of quantum defect theory (QDT) (see, for example, Chang & Fano 1972; Kokoouline & Greene 2005,2003b) using the quantum defect parameters obtained from the ab initio calculation (Mistrik et al. 2001). The constructed scattering matrix accounts for the Jahn-Teller effect and is diagonal with respect to the different irreducible representations $\Gamma$ and the total angular momentum N of the neutral molecule. Thus, the actual calculations are made separately for each $\Gamma$ and N. Elements of the matrix describe the scattering amplitudes for the change of the rovibrational state of the ion after a collision with the electron. However, the S-matrix is not unitary due to the presence of the dissociative vibrational channels (i.e. continuum vibrational states of the ion, discussed above), which are not explicitly listed in the computed S-matrix. The ``defect'' from unitarity of each column of this S-matrix is associated with the dissociation probability of the neutral molecule formed during the scattering process. The dissociation probability per collision is then used to calculate the DR cross-sections and rate coefficients.

The nuclear spin states are characterized by one of the A1, A2, or E irreducible representations of the symmetry group S3 for D3h molecules and by the A or B irreducible representations of the symmetry group S2 for C2v molecules. The irreducible representation  $\Gamma_{\rm ns}$ of a particular nuclear spin state determines its statistical weight and is related to the total nuclear spin $\vec I$ of the state. Here, $\vec I$ is the vector sum of spins $\vec i$ of identical nuclei.

For H3+, the $\Gamma_{\rm ns}=A_1$ states (A2' and A2'' rovibrational states) correspond to I=3/2 (ortho); the $\Gamma_{\rm ns}=E$ states (E' and E'' rovibrational states) correspond to I=1/2 (para). The statistical ortho:para weights are 2:1.

For H2D+, the $\Gamma_{\rm ns}=A$ states (B1 and B2 rovibrational states) correspond to I=1 (ortho); the $\Gamma_{\rm ns}=B$ states (A1 and A2 rovibrational states) correspond to I=0 (para). The statistical ortho:para weights are 3:1.

For D2H+, the $\Gamma_{\rm ns}=A$ states (A1 and A2 rovibrational states) correspond to I=0,2 (ortho); the $\Gamma_{\rm ns}=B$ states (B1 and B2 rovibrational states) correspond to I=1 (para). The statistical ortho:para weights are 2:1.

Finally, for D3+, the $\Gamma_{\rm ns}=A_1$ states (A1' and A1'' rovibrational states) correspond to I=1,3 (ortho); the $\Gamma_{\rm ns}=A_2$ states (A2' and A2'' rovibrational states) correspond to I=0 (para); the $\Gamma_{\rm ns}=E$ states (E' and E'' rovibrational states) correspond to I=1,2 (meta). The statistical ortho:para:meta weights are 10:1:8.

\end{figure} Figure B.1:

Theoretical DR rate coefficients as functions of temperature for the ortho- and para-species of H3+. The figure also shows the species-averaged rate coefficient. For comparison, we show the rate coefficient obtained from measurements in the TSR storage ring by Kreckel et al. (2005) and the analytical dependence for the coefficient used in earlier models of prestellar cores by FPdFW.

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\end{figure} Figure B.2:

Theoretical DR rate coefficients for the ortho- and para-species of H2D+, as functions of temperature. The rate coefficient averaged over the two species and the analytical expression used in earlier models are also shown.

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\end{figure} Figure B.3:

Theoretical DR rate coefficients for the ortho- and para-species of D2H+ as a function of temperature. The rate coefficient averaged over the two species and the analytical expression used in earlier models are also shown.

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\end{figure} Figure B.4:

Theoretical DR rate coefficients for the ortho-, para-, and meta-species of D3+. The rate coefficient averaged over the two species and the one used in earlier models are also shown.

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Figures (B.1)-(B.4) summarize the obtained DR thermal rate coefficients calculated separately for each nuclear spin species of the four H3+ isotopologues and the numerical values are listed in Table B.1. For comparison, the figures also show the analytical dependences used in previous models of prestellar core chemistry (FPdFW). As one can see, the rates for different nuclear spin species are similar to each other (for a given isotopologue) at high temperatures. However, for lower temperatures, the rates for different ortho/para/meta-nuclear spin species significantly differ from each other. The difference in behavior at low temperatures is explained by different energies of Rydberg resonances present in DR cross-sections at low electron energies. The actual energies of such resonances are important for the thermal average at temperatures below or similar to the energy difference between ground rotational levels of different nuclear spin species. At higher temperatures, the exact energy of the resonances is not important. The averaged rate is determined by the density and the widths of the resonances, which are similar for all nuclear spin species over a large range of collision energies.

Table B.1:   Dissociative recombination rates of H3+, H2D+, D2H+, and D3+ for each individual nuclear spin state species.

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